The Grad–Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of first Tokamaks. poloidal surface Tokamak has evolved over years from a circular to D-shaped ellipse. natural geometry that describes such is prolate elliptical one, i.e., cap-cyclide coordinate system. When written this can be terms general Heun function. I...

The classical Heun equation has the form  Q(z) d 2 dz 2 + P (z) d dz + V (z) ff S(z) = 0, where Q(z) is a cubic complex polynomial, P (z) is a polynomial of degree at most 2 and V (z) is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes in [5], [13] initiated the study of the set of all V (z) for which the above equation has a polynomial solution S(z) of a ...

The Schrödinger equation in prolate spheroidal coordinates separates into ordinarydifferential equations in single coordinates; of these direct solutions, two contain confluent Heun functions. We derive polynomial solutions of Heun’s confluent equation and express Coulomb spheroidal functions in a closed algebraic form. Spheroidal orbitals are expressible as hybrids composed of spherical ones c...

We consider a massless, minimally coupled scalar field propagating through the geometry of a black 3-brane in an asymptotically AdS5×S space. The wave equation for modes traveling purely in the holographic direction reduces to a Heun equation and the corresponding greybody factor is obtained numerically. Approximations valid in the lowand high-frequency regimes are also obtained analytically. T...

We investigate the structure of Wigner distribution functions energy eigenstates quartic and sextic anharmonic oscillators. The corresponding Moyal equations are shown to be solvable, revealing new properties Schrödinger eigenfunctions these In particular, they provide alternative approaches their determination eigenenergies, away from conventional differential equation method, which usually re...

We give explicit expressions for the finite frequency greybody factor, quasinormal modes, and Love numbers of Kerr black holes by computing exact connection coefficients radial angular parts Teukolsky equation. This is obtained solving problem confluent Heun equation in terms expression irregular Virasoro conformal blocks as sums over partitions via Alday, Gaiotto, Tachikawa correspondence. In ...

In this paper, we consider the monodromy and, in particular, isomonodromy sets of accessory parameters for Heun class equations. We show that equations can be obtained as limits linear systems associated with Painlevé when transcendents go to one actual singular points systems. The are described by Taylor or Laurent coefficients corresponding functions, tau at positions critical values. As an a...

In this paper, we investigate the Hankel determinant generated by a singular Laguerre weight with two parameters. Using ladder operators adapted to monic orthogonal polynomials associated weight, show that one of auxiliary quantities is solution Painlevé III′ equation and derive discrete σ-forms logarithmic partial derivatives determinant. We approximate second-order differential satisfied resp...